P.K. Rajan

2-D symmetry: theory and filter design applications

In this comprehensive review article, we present the theory of symmetry in two-dimensional (2-D) filter functions and in 2-D Fourier transforms. It is shown that when a filter frequency response possesses symmetry, the realization problem becomes relatively simple. Further, when the frequency response has no symmetry, there is a technique to decompose that frequency response into components each of which has the desired symmetry. This again reduces the complexity of two-dimensional filter design. A number of filter design examples are illustrated.

Symmetry study on 2-D complex analog and digital filter functions

The symmetry properties of 2-D complex polynomials are analyzed in this paper. The characteristics of a polynomial possessing different standard symmetries in their magnitude and phase responses are studied. The nature of constraints that are imposed by the defined symmetries on analog and digital polynomials is discussed. The various classes of complex polynomials possessing the different (quadrantal, diagonal, rotational, and octagonal) symmetries and antisymmetries in their magnitude responses and/or phase responses are tabulated.

Independent component analysis of digital image watermarking

In this paper independent component analysis (ICA) is used to test the robustness of watermarked image. By applying suitable ICA forward transformation, the distributions of the cover image and the watermark are modified. The cover image and the watermark are estimated by using a nonlinear filter and the inverse transform matrix. The preliminary experiments show that ICA attack not only influences the hidden information but also improves the image quality for reasonable embedding strengths.

A study of two-variable very strict Hurwitz polynomials with interval coefficients

The derivation of Kharitonov-type extremum polynomials for two-variable very strict Hurwitz interval polynomials using circuit theoretic techniques is presented. In this connection very strict reactance functions are defined and their properties studied. Applying the properties of two-variable very strict Hurwitz polynomials, extremum positive functions are identified to test interval positive functions belonging to the very strict category.<<ETX>>

Least square error optimal 2-D FIR filter design for nonsymmetric and symmetric specifications in discrete frequency domain circuits

The least square error optimal design of two-dimensional FIR (finite impulse response) digital filters when the specifications are given in the discrete frequency domain is examined in detail. Employing Parsevals relation for 2-D DFT (discrete Fourier transform) arrays, optimal filter impulse response is determined for the filter coefficients. Simplified expressions for the cases where the frequency response possesses quadrantal symmetry are also derived.<<ETX>>

Symmetry conditions for 2-D complex irreducible polynomials

Constraints on analog and digital complex irreducible polynomials are given, such that they possess different types of symmetries, such as centrosymmetry, quadrantal symmetry, diagonal symmetry, four-fold rotational symmetry, and octagonal symmetry, in their magnitude responses. The symmetries present in the phase responses of these polynomials are also discussed. It is further shown that the 3-D single planar symmetry results for real polynomials have special relations to the 2-D symmetry results for complex polynomials, and that the 2-D symmetry results for 2-D real polynomials are special cases of those for the complex polynomials.<<ETX>>

Unified theory of symmetry for two-dimensional complex polynomials using delta discrete-time operator

The complexity in the design and implementation of 2-D filters can be reduced considerably if the symmetries that might be present in the frequency responses of these filters are utilized. As the delta operator (γ-domain) formulation of digital filters offers better numerical accuracy and lower coefficient sensitivity in narrow-band filter designs when compared to the traditional shift-operator formulation, it is desirable to have efficient design and implementation techniques in γ-domain which utilize the various symmetries in the filter specifications. Furthermore, with the delta operator formulation, the discrete-time systems and results converge to their continuous-time counterparts as the sampling periods tend to zero. So a unifying theory can be established for both discrete- and continuous-time systems using the delta operator approach. With these motivations, we comprehensively establish the unifying symmetry theory for delta-operator formulated discrete-time complex-coefficient 2-D polynomials and functions, arising out of the many types of symmetries in their magnitude responses. The derived symmetry results merge with the s-domain results when the sampling periods tend to zero, and are more general than the real-coefficient results presented earlier. An example is provided to illustrate the use of the symmetry constraints in the design of a 2-D IIR filter with complex coefficients. For the narrow-band filter in the example, it can be seen that the γ-domain transfer function possesses better sensitivity to coefficient rounding than the z-domain counterpart.

Efficient design of delta operator based 2-D IIR filters using symmetrical decomposition

A method based on symmetrical decomposition is presented for the design of 2-D IIR filters through optimization. This technique allows a filter with non-symmetric magnitude specification to benefit from the savings in parameters provided by symmetry. The procedure is developed for the delta operator based design to take advantage of its better coefficient sensitivity compared to the z-domain design when the filter is narrowband.

Generalized /spl alpha/-VSH polynomials and stability of /spl delta/-operator based 2D discrete-time systems

Delta (/spl delta/-) operator formulation for discrete time systems with sampling period T as explicit variable merges to the underlying continuous time system as T/spl rarr/0. The paper uses this fact in studying the stability of two-dimensional (2D) /spl delta/-discrete time (DT) systems and formulates the concept of the /spl alpha/-very strict Hurwitz polynomial (VSHP). This is related to the 2D (1//spl alpha/)-Schur polynomial through inverse variable transformation. The properties and test procedures for /spl alpha/-VSHF and its importance in the structural stability of 2D /spl delta/-DT systems are presented.

Study of the compatibility of 1-D and 2-D bounded real functions

Recently Zhu and Chen presented a unified theory concerning the compatibility of two impedances. The authors present an alternate simplified formulation to the compatibility problem using Belevitchs canonical scattering parameter representation for two-ports. The problem is addressed in terms of compatibility of two bounded real-functions, /spl rho//sub 1/(s) and /spl rho//sub 2/(s). It is established that the crux of the problem is identifying an auxiliary polynomial /spl lambda/ such that a certain polynomial involving /spl lambda/ is strictly Hurwitz or scattering Hurwitz after cancellation of the factors involving the transmission zeros. Although results are presented in the analog network, with suitable modification, the results could be formulated in a discrete time network. When transmission zeros associated with /spl rho//sub 2/(s) are all at s = /spl infin//0 it is necessary to test for Hurwitz property of a polynomial and there is no need to find the auxiliary polynomial /spl lambda/.<<ETX>>